Consider the GARCH(1,1)model $\sigma _ { t } ^ { 2 }$

Essay

Consider the GARCH(1,1)model $\sigma _ { t } ^ { 2 }$ = ?₀ + ?₁
$u _ { t - 1 } ^ { 2 }$ + ?₁
$\sigma _ { t - 1 } ^ { 2 }$ Show that this model can be rewritten as $\sigma _ { t } ^ { 2 }$ = $\frac { \alpha _ { 0 } } { 1 - \phi _ { 1 } }$ + ?₁( $u _ { t - 1 } ^ { 2 }$ + ?₁
$u _ { t - 2 } ^ { 2 }$ + $\phi _ { 1 } ^ { 2 }$ $u _ { t - 3 } ^ { 2 }$ + $\phi _ { 1 } ^ { 3 }$ $u _ { t - 4 } ^ { 2 }$ + ...). (Hint: use the GARCH(1,1)model but specify it for $\phi _ { t - 1 } ^ { 2 }$ ; substitute this expression into the original specification, and so on.)Explain intuitively the meaning of the resulting formulation.